Chapter 19, Problem 27E

### Chemistry

9th Edition
Steven S. Zumdahl
ISBN: 9781133611097

Chapter
Section

### Chemistry

9th Edition
Steven S. Zumdahl
ISBN: 9781133611097
Textbook Problem

# Krypton consists of several radioactive isotopes, some of which are listed in the following table.   Half-Life 73Kr 27 s 74Kr 11.5 min 76Kr 14.8 h 81Kr 2.1 X 105 yr Which of these isotopes is most stable, and which isotope is “hottest”? How long does it take for 87.5% of each isotope to decay?

Interpretation Introduction

Interpretation: List of isotopes of krypton is given. The most stable and the hottest among them is to be stated. Time of decay of 87.5% of each isotope is to be stated.

Concept introduction: Decay constant is the quantity that expresses the rate of decrease of number of atoms of a radioactive element per second. Half life of radioactive sample is defined as the time required for the number of nuclides to reach half of the original value.

The nuclides having longer half life are more stable while nuclides having shorter half life are less stable.

To determine: The most stable and the hottest isotope among the given isotopes of krypton; the time of decay for 73Kr ; the time of decay for 74Kr ; the time of decay for 76Kr and the time of decay for 81Kr .

Explanation

Explanation

The most stable isotope is 81Kr and the hottest one is 73Kr

The nuclides having longer half life are more stable while nuclides having shorter half life are less stable. Thus the most stable isotope is 81Kr and the hottest isotope is 73Kr .

The time of decay for 73Kr is 81.428s_

Explanation

The decay constant can be calculated by the formula given below.

λ=0.693t1/2

Where

• t1/2 is the half life of nuclide.
• λ is the decay constant.

Substitute the value of λ in the above expression.

λ=0.69327s1=0.0256s1

The fraction of isotope decayed is 87.5100 .

The fraction remaining =187.5100=12.5100

The time of decay can be calculated by the formula,

t=2.303λlogn0n

Where

• n0 is the number of atoms initially present.
• n is the number of atoms after time “t”.

Substitute the values of λ , n0 and n in the above expression.

t=2.303λlogn0nt=2.303λlog10012.5t=2.3030.0256log10012.5t=81.428s_ .

The time of decay for 74Kr is 34.51s_ .

Explanation

The decay constant is calculated by the formula,

λ=0.693t1/2

Where

• t1/2 is the half life of nuclide.
• λ is the decay constant.

Substitute the value of t1/2 in the above expression.

λ=0.69311.5min1

The fraction of isotope decayed is 87.5100 .

The fraction remaining =187.5100=12.5100

The time of decay can be calculated by the formula,

t=2.303λlogn0n

Where

• n0 is the number of atoms initially present.
• n is the number of atoms after time “t”.

Substitute the values of λ , n0 and n in the above expression.

t=2.303λlogn0nt=2.303λlog10012.5t=2.303×11.50.693log8t=34.51min_

The time of decay for 76Kr is 44

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